What are the divisors of 259?

1, 7, 37, 259

There is a total of 4 positive divisors.
The sum of these divisors is 304.
The arithmetic mean is 76.

4 odd divisors

1, 7, 37, 259

How to compute the divisors of 259?

A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 259 by each of the numbers from 1 to 259 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

259 / 1 = 259 (the remainder is 0, so 1 is a divisor of 259)
259 / 2 = 129.5 (the remainder is 1, so 2 is not a divisor of 259)
259 / 3 = 86.333333333333 (the remainder is 1, so 3 is not a divisor of 259)
...
259 / 258 = 1.0038759689922 (the remainder is 1, so 258 is not a divisor of 259)
259 / 259 = 1 (the remainder is 0, so 259 is a divisor of 259)

Improved algorithm using square-root

However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 259 (i.e. 16.093476939431). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

259 / 1 = 259 (the remainder is 0, so 1 and 259 are divisors of 259)
259 / 2 = 129.5 (the remainder is 1, so 2 is not a divisor of 259)
259 / 3 = 86.333333333333 (the remainder is 1, so 3 is not a divisor of 259)
...
259 / 15 = 17.266666666667 (the remainder is 4, so 15 is not a divisor of 259)
259 / 16 = 16.1875 (the remainder is 3, so 16 is not a divisor of 259)